1. Fibonacci Property Proof help

Hi,
I was hoping that someone could help me with some proofs for the Fibonacci sequence.

1) If p is prime then $F_p \equiv \ \frac{5}{p} (mod p)$
2) If p is prime then $2F_{p+1} \equiv \ 1+ \frac{5}{p} (mod p)$

Thanks!

2. Hmm?

2 is prime. $F2 = 1 \equiv 1 ~ (mod ~ 2)$
3 is prime. $F3 = 1+1 = 2 \equiv 2 ~ (mod ~ 3)$

Are you sure this is the intended question?

3. I took the equation directly from the sheet because I am having the same difficulties and currently just completely confused. I should have mentioned this in the first post but it is going off of the assumption that $F_1 = 1 , F_2 = 1 , F_3=2, F_4=3,..., F_n = F_{n-1} + F_{n-2}$ for $n \geq \ 3$ So I don't know if that changes anything.

4. Isn't it supposed to be: $F_p\equiv{\left(\frac{5}{p}\right)_L}(\bmod.p)$ where $\left(.\right)_L$ is the Legendre Symbol ?

I give you the ingredients: Binet's Formula,the Binomial Theorem, Fermat's Little Theorem and Euler's Criterion.

5. Yes I think you are right about it being a Legendre Symbol.